Question

A wire stretched between two rigid supports vibrates in its fundamental mode with a frequency of 45 Hz. The mass of the wire is $3.5\times {10}^{-2}kg$ and its linear mass density is $4.0\times {10}^{-2}kg{m}^{-1}$. What is (a) the speed of a transverse wave on the string, and (b) the tension in the string ?

Answer 1

(a) Mass of the wire, $m=3.5\times {10}^{-2}kg$

Linear mass density, $\mu =m/l=4.0\times {10}^{-2}kg{m}^{-1}$

Frequency of vibration, $f=45Hz$

$\therefore$ Length of the wire, $l=\frac{m}{\mu }=\frac{3.5\times {10}^{-2}}{4.0\times {10}^{-2}}=0.875m$

The wavelength of the stationary wave (λ) is related to the length of the wire by the relation:

$\lambda =2l/n$
where,
$n=$ Number of nodes in the wire
For fundamental node, $n=1$:

$\lambda =2l$

$\lambda =2\times 0.875=1.75m$
The speed of the transverse wave in the string is given as:

$v=f\lambda =45\times 1.75=78.75m/s$
(b) The tension produced in the string is given by the relation:

$T=v^2\mu$

$=(78.75{)}^2\times 4.0\times {10}^{-2}=248.06N$